*2.1. Basic Equations*

The unsteady two-dimensional flow of micropolar Cu–Al2O3/water nanofluid past a deformable sheet in the stagnation region with the influence of thermal radiation impact are investigated in this work as exemplified in Figure 1. The Cartesian coordinates used are *x* and *y*, given that *<sup>x</sup>*−axis is considered along the sheet while *y*−axis normal to it, respectively, the sheet is located in the plane *y* = 0 and the fluid fill the half space at *y* ≥ 0. The temperature far from the surface (inviscid flow) and at the surface are represented by *T*∞ and *Tw*(*<sup>x</sup>*, *t*). The sheet is stretch and shrunk along the *<sup>x</sup>*−axis with velocity *uw*(*<sup>x</sup>*, *t*) and the free stream velocity is denoted by *ue*(*<sup>x</sup>*, *t*).

**Figure 1.** Schematic model of shrinking sheet.

From all of the above circumstance, the partial differential equations which govern the flow are stated as (see Nazar et al. [9], Bhattacharyya et al. [54], Roy et al. [55]):

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \tag{1}$$

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \frac{\partial u\_\varepsilon}{\partial t} + u\_\varepsilon \frac{\partial u\_\varepsilon}{\partial x} + \frac{\mu\_{huf} + \kappa}{\rho\_{huf}} \frac{\partial^2 u}{\partial y^2} + \frac{\kappa}{\rho\_{huf}} \frac{\partial N}{\partial y} \tag{2}$$

$$
\rho \frac{\partial N}{\partial t} + \mu \frac{\partial N}{\partial x} + v \frac{\partial N}{\partial y} = \frac{\varrho}{\rho\_{\rm Imf} \dot{f}} \frac{\partial^2 N}{\partial y^2} - \frac{\kappa}{\rho\_{\rm Imf} \dot{f}} \left(2N + \frac{\partial u}{\partial y}\right) \tag{3}
$$

$$\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \frac{k\_{\text{huf}}}{\left(\rho \mathbb{C}\_p\right)\_{\text{huf}}} \frac{\partial^2 T}{\partial y^2} - \frac{1}{\left(\rho \mathbb{C}\_p\right)\_{\text{huf}}} \frac{\partial q\_r}{\partial y} \tag{4}$$

Here, the velocity component in the *x* direction is denoted as *u* whereas *v* is the velocity component along *y* axis, *t* and *T* are time and temperature, *N* refers to the angular velocity (microrotation) in the *xy*−plane, *qr* signifies the radiative heat flux, *κ* is the vortex viscosity and *j* is the micro inertial density. In addition, *ς* is the spin gradient viscosity given by (Ahmadi [6])

$$
\xi = \left(\mu\_f + \frac{\kappa}{2}\right)\dot{\jmath} \tag{5}
$$

where *j* = *νf* /*ue* is specified as the reference length. Further, *khnf* , *ρhnf* , *μhnf* and *<sup>ρ</sup>Cp hnf* are the thermal conductivity, density, dynamic viscosity, and heat capacity of Cu–Al2O3/ water.

The accompanying conditions are

$$\begin{aligned} \mu = \mu\_w(\mathbf{x}, t), \; v = 0, \; N = -n \frac{\partial \mu}{\partial y'}, \; T = T\_w(\mathbf{x}, t) \text{ as } y = 0\\ \mu \to \mu\_\varepsilon(\mathbf{x}, t), \; N \to 0, \; T \to T\_\infty \text{ as } y \to \infty \end{aligned} \tag{6}$$

where *n* is the constant in the range of [0, 1]. It is worthwhile to note that for *n* = 0 which implies that *N* = 0 near the wall, exemplifies the microelements near the wall surface are incapable to rotate, i.e., concentrated particle flows (Jena and Mathur [56]) or also denoted as strong concentration of microelements (Guram and Smith [57]). However, for the case *n* = 0.5 which refer to a weak concentration of microelements, the disappearing of anti-symmetric part of the stress tensor is noted (Ahmadi [6]). Further, the case *n* = 1 is utilized for the modelling of turbulent boundary layer flows (Peddieson [58]). While the velocity of deformable sheet, free stream and temperature at the surface are referred from the work of Zainal et al. [59] which given as

$$u\_w(\mathbf{x},t) = \frac{c\mathbf{x}}{1 - bt},\ u\_t(\mathbf{x},t) = \frac{a\mathbf{x}}{1 - bt},\ T\_w(\mathbf{x},t) = T\_\infty + \frac{T\_0 a\mathbf{x}^2}{2\nu\_f(1 - bt)^{3/2}}\tag{7}$$

here, *a*(<sup>&</sup>gt; 0) and *c*(<sup>&</sup>gt; 0) are constants, *b* measures the unsteadiness of the problem and *T*0 > 0 is the reference temperature.

Using the Rosseland's approximation (Brewster [60]), the *qr* term can be expressed clearly as below

$$\eta\_I = -\frac{4}{3} \frac{\sigma^\*}{k^\*} \frac{\partial T^4}{\partial y} \tag{8}$$

where *σ*<sup>∗</sup> and *k*∗ signify the constant of Stefan–Boltzmann and mean absorption's coefficient. Implementing the Taylor series and ignored the higher-order terms, *T*<sup>4</sup> is expanded about *T*∞; hence, we have *T*<sup>4</sup> ≈ <sup>4</sup>*T*3∞*<sup>T</sup>* − <sup>3</sup>*T*4∞. Subsequently, Equation (4) become

$$\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial \mathbf{x}} + v \frac{\partial T}{\partial y} = \frac{k\_{huf}}{\left(\rho \mathbb{C}\_p\right)\_{hm}} \frac{\partial^2 T}{\partial y^2} + \frac{16 \,\sigma^\* \, T\_{\infty}^3}{3 \, k^\* \left(\rho \mathbb{C}\_p\right)\_{hmf}} \frac{\partial^2 T}{\partial y^2} \tag{9}$$
